#2588

Count the Number of Beautiful Subarrays

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array · hash table · bit manipulation · prefix sum

Description

You are given a 0-indexed integer array nums. In one operation, you can:

- Choose two different indices i and j such that 0 <= i, j < nums.length.

- Choose a non-negative integer k such that the k^th bit (0-indexed) in the binary representation of nums[i] and nums[j] is 1.

- Subtract 2^k from nums[i] and nums[j].

A subarray is beautiful if it is possible to make all of its elements equal to 0 after applying the above operation any number of times (including zero).

Return the number of beautiful subarrays in the array nums.

A subarray is a contiguous non-empty sequence of elements within an array.

Note: Subarrays where all elements are initially 0 are considered beautiful, as no operation is needed.

Example 1:

Input: nums = [4,3,1,2,4]

Output: 2

Explanation: There are 2 beautiful subarrays in nums: [4,3,1,2,4] and [4,3,1,2,4].

- We can make all elements in the subarray [3,1,2] equal to 0 in the following way:

- Choose [3, 1, 2] and k = 1. Subtract 21 from both numbers. The subarray becomes [1, 1, 0].

- Choose [1, 1, 0] and k = 0. Subtract 20 from both numbers. The subarray becomes [0, 0, 0].

- We can make all elements in the subarray [4,3,1,2,4] equal to 0 in the following way:

- Choose [4, 3, 1, 2, 4] and k = 2. Subtract 22 from both numbers. The subarray becomes [0, 3, 1, 2, 0].

- Choose [0, 3, 1, 2, 0] and k = 0. Subtract 20 from both numbers. The subarray becomes [0, 2, 0, 2, 0].

- Choose [0, 2, 0, 2, 0] and k = 1. Subtract 21 from both numbers. The subarray becomes [0, 0, 0, 0, 0].

Example 2:

Input: nums = [1,10,4]

Output: 0

Explanation: There are no beautiful subarrays in nums.

Solution